Adaptive Tracking Control of Uncertain MIMO Nonlinear Systems with Time-varying Delays and Unmodeled Dynamics

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1 Internatonal Journal of Automaton and Computng (3), June 3, 94- DOI:.7/s Adaptve Trackng Control of Uncertan MIMO Nonlnear Systems wth Tme-varyng Delays and Unmodeled Dynamcs Xao-Cheng Sh Tan-Png Zhang Department of Automaton, College of Informaton Engneerng, Yangzhou Unversty, Yangzhou 57, Chna Abstract: In ths paper, adaptve neural trackng control s proposed based on radal bass functon neural networks (RBFNNs) for a class of mult-nput mult-output (MIMO) nonlnear systems wth completely unknown control drectons, unknown dynamc dsturbances, unmodeled dynamcs, and uncertantes wth tme-varyng delay. Usng the Nussbaum functon propertes, the unknown control drectons are dealt wth. By constructng approprate Lyapunov-Krasovsk functonals, the unknown upper bound functons of the tme-varyng delay uncertantes are compensated. The proposed control scheme does not need to calculate the ntegral of the delayed state functons. Usng Young s nequalty and RBFNNs, the assumpton of unmodeled dynamcs s relaxed. By theoretcal analyss, the closed-loop control system s proved to be sem-globally unformly ultmately bounded. Keywords: Adaptve control, unmodeled dynamcs, tme-varyng delays, neural networks, Nussbaum functon Introducton Durng the past several decades, the desgn of adaptve controller for nonlnear systems has receved much attenton based on neural network (NN) approxmaton performance. Several stable adaptve NN control approaches were nvestgated [ 3]. But there exst some types of matchng condtons n most of these NN control schemes. Attemptng to overcome these restrctons, adaptve backsteppng desgn was presented for a class of strct-feedback nonlnear systems usng neural network control [4]. Research on adaptve control technques for mult-nput mult-output (MIMO) nonlnear systems s of great sgnfcance. Usng multrate sampled-data control technque, an adaptve output feedback control was proposed for a class of general MIMO systems [5]. Based on the prncple of sldng mode control and usng the propertes of Nussbaum-type functon, several adaptve neural control schemes were presented for a class of uncertan nonlnear state tme delay systems [6 8]. In addton, tme delay uncertantes were compensated for by constructng approprate Lyapunov- Krasovsk functonals n them. It s well known that tme delays are often encountered n varous systems, and the exstence of tme delays usually becomes the source of nstablty and makes the control synthess more dffcult. Lyapunov-Krasovsk functonal n [9] and Lyapunov- Razumkhn functonal n [] are essental tools n the stablty analyss of tme-delay systems. Recently, several adaptve control schemes have been proposed that combned the Lyapunov-Krasovsk functonal and the radal bass functon neural networks (RBFNNs) or fuzzy logc systems for nonlnear systems wth tme delays [ 6]. In [ 3], three adaptve control schemes were developed for a class of nonlnear systems wth constant delays. But tme-varyng delays often appeared n many systems [4 7]. Manuscrpt receved September, ; revsed December 4, Ths work was supported by Natonal Natural Scence Foundaton of Chna (No ). Based on the prncple of sldng mode control and usng the ntegral-type Lyapunov functon, adaptve control desgn was nvestgated for a class of uncertan MIMO state tmevaryng delay nonlnear systems wth unknown dsturbances and known gan sgns [4]. In [5], by ntroducng Max functon to construct contnuous approxmaton functon, a class of MIMO nonlnear systems wth unknown dead-zone and tme-varyng delays was studed by varable structure control. Combnng the separaton technque wth the dynamc surface control approach, and constructng an approprate Lyapunov-Krasovsk functonal, an adaptve dynamc surface control desgn was proposed for a class of nonlnear systems wth unknown dscrete and dstrbuted tme-varyng delays [6]. Usng the parameter separaton technque and the Nussbaum-type functon propertes, a novel adaptve teratve learnng control approach was proposed for a class of nonlnearly parameterzed systems wth unknown tmevaryng delay and unknown control drecton [7]. As we all known, unmodeled dynamcs exsts n many practcal nonlnear systems, due to many factors, such as measurement nose, modelng errors, external dsturbances, modelng smplfcatons, and they can severely degrade the closed-loop system performance. Therefore, varous approaches were nvestgated to handle such systems wth unmodeled dynamcs [8 ]. In [8], a recursve robust adaptve control was proposed for a class of nonlnear systems wth unmodeled dynamcs by applyng nput-to-state stablty property. However, the control gan and ts drecton are assumed to be known n the consdered systems. Usng Nussbaum-type functons and fuzzy logc systems, a drect adaptve robust control was presented for a class of snglenput sngle-output (SISO) nonlnear systems wth unmodeled dynamcs and unknown vrtual control drectons. The dea n [8, 9] was to use an avalable dynamc sgnal to conduct unmodeled dynamcs. Combnng backsteppng wth dynamc surface control, a novel adaptve control was presented for a class of nonlnear systems n pure feed-

2 X. C. Sh and T. P. Zhang / Adaptve Trackng Control of Uncertan MIMO Nonlnear Systems 95 back form wth unmodeled dynamcs []. The assumptons wth respect to unmodeled dynamcs [] are smlar to the work n [] whch satsfy a so-called lower-trangular condton. Tong et al. [8 ] only consdered unmodeled dynamcs alone, but n practce, unmodeled dynamcs and tmevaryng delays often appear smultaneously. Motvated by the prevous works, adaptve neural network control s nvestgated for a class of MIMO nonlnear systems n ths paper. The man contrbutons of the paper are as follows. ) Adaptve neural trackng control s proposed for a class of MIMO nonlnear systems wth unmodeled dynamcs, unknown dynamc dsturbances, completely unknown control drectons, and tme-varyng delay uncertantes. To the best of our knowledge, t s the frst tme to deal wth such knd of systems wth several uncertantes as compared to the exstng lteratures. ) By constructng the approprate Lyapunov-Krasovsk functonals n the control desgn, the assumptons wth respect to the unmodeled dynamcs and the dynamc dsturbances as well as the tme-varyng delays are relaxed [6 8,, 4, 5,, ]. The upper bounds of the tme-varyng delay and the dynamc dsturbance may be unknown. Moreover, the controller desgn does not calculate the tme-varyng gan usng the functons of the upper bound of the tme-varyng delayed uncertantes [6 8,, 4, 5]. Ths paper s organzed as follows. The problem formulaton and prelmnares are gven n Secton. In Secton 3, an adaptve RBFNN control s developed for a class of MIMO nonlnear systems by utlzng an ntegral-type Lyapunov functon. Furthermore, the closed-loop system stablty s analyzed as well. In Secton 4, smulaton results are demonstrated. Fnally, Secton 5 detals the conclusons. Problem formulaton and prelmnares. Problem formulaton Consder a class of uncertan MIMO nonlnear systems wth unmodeled dynamcs and tme-varyng delay n the followng form: ż = q(t, z, x) ẋ = x,, =,,n ẋ n = f (x)h (x τ )b (x, x,, x m)u Δ (t, z, x) ẋ = x,, =,,n ẋ n = f (x, u,,u )b ( x, x,, x m)u h (x τ)δ (t, z, x),=, 3,,m y = x,,y m = x m () where z R n s the unmodeled dynamcs, x = [x T,x T,,x T m] T R n s the state vector, x = [x,,x n ] T, =,,m, n = m = n, u R, y R are the -th system s nput and output, x = [x T,x T,,x T ] T, x = [x,,x,n ] T, x τ = [x T (t τ (t)),x T (t τ (t)),,x T m(t τ m(t))] T, τ (t), =,,,m, are the unknown tme-varyng delays, h (x τ), f (x, u,,u ) are the unknown contnuous functons, b ( x, x,, x m) are the unknown dfferentable control gans, and Δ (t, z, x) denotes the unknown uncertan dsturbance. For t [ τ max, ], x(t) =ω(t), ω(t) saknown contnuous ntal state vector functon. τ max, whch wll be defned later, s an unknown postve constant. The control obectve s to desgn control u for system () such that the output y follows the desred traectory y d,=,,,m. Defne x d, e and s as x d =[y d, ẏ d,,y (n ) d ] T e = x x d =[e,e,,e n ] T s =( d dt λ)n e = n c e e n () where c = C n λn, =,,,n,λ >, =,,,mare postve constants, specfed by the desgner. Assumpton. The desred traectory vectors are contnuous and avalable, and x d = [x T d,y (n ) d ] T Ω d R n,ω d s a known compact, =,,,m. Assumpton. There exst unknown postve contnuous functons ρ k (x k (t)) and unknown postve constants φ such as h (x τ ) φ m k= ρ k(x k (t τ k (t))), =,,,m. Assumpton 3. The unknown state tme-varyng delays τ (t) satsfy τ (t) τ max, τ (t) τ max <, =,,,m, wth the unknown constants τ max and τ max. Assumpton 4. Smooth functons b ( x, x,, x m) and ther sgns are unknown, and there exst constants b and b such that <b b ( x, x,, x m) b, =,,m. Assumpton 5. The equlbrum z = of system ż = q(t,z, ) q(t,, ) s globally asymptotcally stable,.e., there exsts a Lyapunov functon W satsfyng the followng nequaltes c z W (t, z) c z W t (t, z) W z c 3 z W z (t, z)(q(t, z, ) q(t,, )) (t, z) c4 z where c,c,c 3, and c 4 are unknown postve constants, denotes Eucldean norm. Also, there exsts an unknown postve constant c 5 such that q(t,, ) c 5, t. Assumpton 6. There exst unknown postve constant p and unknown contnuous functon ψ C, ψ () = such that q(t, z, x) q(t, z, ) p ψ ( x ). Assumpton 7. There exst unknown postve constants, p, such that (x, t) R n R Δ (t, z, x) p ρ ( x )p z ρ (x) where ρ ( x ) andρ (x) are unknown postve contnuous functons, =,,m.

3 96 Internatonal Journal of Automaton and Computng (3), June 3 Remark. The functons ρ ( x ) andρ (x) are unknown n ths paper whle they are known n [, ]. In addton, ψ ( x ) contans all the state varables. Compared wth the exstng lteratures [, ], the assumptons wth respect to dynamc dsturbances and unmodeled dynamcs are relaxed. Lemma. (Young s nequalty [] ) For vectors x, y R n, the followng nequalty holds. x T y εp x p p y q qε q where ε>,p>,q >, and (p )(q ) =.. Radal bass functon neural networks In ths paper, we wll employ radal bass functon (RBF) neural networks to approxmate unknown contnuous functon h(z): R q R as dscussed n [3, 4]. Then, we have h(z) =W T S(Z)δ(Z) (3) where S(Z) = [s (Z),s (Z),,s l (Z)] T R l s the bass functon vector, s (Z) = e (Z µ )T(Z µ ) k, =,,,l, μ =[μ,μ,,μ q] T s the center of the receptve feld, k s the wdth of the Gaussan functon, and W s an unknown deal constant weght vector. It s defned as W = arg mn sup { W T S(Z) h(z) }, where W R l Z Ω Z δ(z) s the approxmaton error, satsfyng δ(z) ε wth ε beng an unknown postve constant. Choose λ = W rather than W as the parameter. ˆλ s the estmaton of the unknown constants λ,andthe estmaton error satsfes λ = λ ˆλ..3 Nussbaum functon propertes In order to deal wth the unknown control gan sgn, the Nussbaum gan technque s employed n ths paper. A functon N(ς) s called Nussbaum-type functon f t has the followng propertes: ) lm sup s N(ς)dς =, s s ) lm nf s N(ς)dς =. s s In ths paper, N(ς) =e ς cos( πς )sused. Lemma [8]. Let V ( ) and ς( ) be smooth functons defned on [,t f )wthv(t), t [,t f ), and N( ) be an even smooth Nussbaum-type functon. If the followng nequalty holds: V (t) c e c t g(x(τ))n(ς) ςe cτ dτ e c t ςe cτ dτ, t [,t f ) (4) where c represents some sutable constant, c s a postve constant, and g(x(τ)) s a tme-varyng parameter whch takes values n the unknown closed ntervals I =[l,l ], wth / I, thenv (t),ς(t) and g(x(τ))n(ς) ς dτ must be bounded on [,t f ). 3 Control system desgn and stablty analyss Consder the -th subsystem, we obtan ṡ = γ f (x, u,,u )h (x τ) b ( x, x,, x m)u Δ (t, z, x) (5) where γ = n ce, y(n ) d. Defne a smooth scalar functon as V s = s σ b ( x, x,σ β, x,, dσ (6) x m) where β = y (n ) d n ce. Obvously, we can conclude β = γ. By second mean value theorem for ntegrals, we can obtan V s = s b ( x, x,λs s β, x,, x m) wth λ s (, ). Because <b b ( x, x,, x m), t s shown that V s s postve defnte wth respect to s. Let V w = λ W,whereλ s a postve constant, and W s gven n Assumpton. Dfferentatng V w wth respect to tme t, wehave V w = [ W λ t W ż] z [ c 3 z c 4c 5 z c 4p λ z ψ ( x )] c 3 z c 3 z λ c 4c 5 λ 8λ c 3 c 3 z λc 4p ψ ( x ) 8λ c 3 c 3 z c 3 z λ c λ 8λ c 4c 5 3 c 3 z λ c 4 4p 4 ψ 8λ ( x ). 4 For the sake of clarty and convenence, let c 3 B = b ( x, x,, x m) F (σ, β )= b ( x, x,σ β, x,, x m) G,k(σ, β )= b ( x, x,σ β, x,, x m). x,k Dfferentatng (6) wth respect to tme t, weobtan V s = s B ṡ s β s m n k= s σg,n (σ, β )ẋ,n dσ σg,k(σ, β )x,k dσ σ F(σ, β) β dσ. (7)

4 X. C. Sh and T. P. Zhang / Adaptve Trackng Control of Uncertan MIMO Nonlnear Systems 97 Substtutng (5) nto (7) and usng β = γ,weobtan where V s = s (f (x, u,,u ) B h (x τ)b ( x, x,, x m)u Δ (t, z, x)) m n k= s s σg,k(σ, β )x,k dσ s γ F (σ, β )dσ σg,n (σ, β )[f (x, u,,u )h (x τ ) b ( x, x,, x m)u Δ (t, z, x)]dσ. (8) Applyng Young s nequalty, Assumptons and 5, we obtan s B h (x τ) s m k= φ s B m ρ k(x k (t τ k (t))) (9) k= σg,n (σ, β )h (x τ)dσ s Δ (t, z, x) B s m ( k= k= s σg,n (σ, β )dσ) φ m ρ k(x k (t τ k (t))) () s ρ ( x ) a B c 3 z 4λ b p 4 a p s4 ρ 4 (x) b B4 G,n (σ, β )σδ (t, z, x)dσ s ( s ( σg,n (σ, β )dσ) p a c3 z 4λ ρ ( x ) a b p 4 σg,n (σ, β )dσ) 4 ρ 4 (x)λ b c 3 () () g (t) = b( x, x,, x m) b ( x, x,, x m) ϕ (Z )= s 3 ρ 4 (x) b B4 f(x, u,,u ) B s m n k= γ F (σ, β )dθ s m k= φ s B sρ ( x ) a B G,k(σ, β )θx,k dθ θg,n (σ, β ) [f (x, u,,u )b ( x, x,, x m)u ]dθ m [ ] G,n (σ, β )θdθ φ s 3 k= ρ ( x )s 3 a [ θg,n (σ, β )dθ] [ ] 4 θg ρ 4 (x)λ s 7,n (σ, β )dθ b c 3 Z =[x T,s,γ,β,u,,u ] T Ω Z. (4) To overcome the desgn dffcultes from the unknown tme delay n (3), consder the followng Lyapunov- Krasovsk functonal V U = e γ(t τmax) ( τ max) m k= t τ k (t) e γτ ρ k(x k (τ))dτ. (5) Dfferentatng (5) wth respect to t and usng (3) we have V s V w V U s (g (t)u h (Z )) c3 z 4λ ( s c )ψ 4 ( x ) e γτmax (c s ) ( τ max)c k= a p b p 4 m ρ k(x k (t)) γ VU (6) where a,b are postve constants. Substtutng (9) () nto (8) yelds V s V w s g (t)u s ϕ (Z ) m ρ k(x k (t τ k (t))) c3 z 4λ k= ψ 4 ( x ) a p b p 4 (3) where eγτmax s h (Z )=ϕ (Z ) ( τ max)c s c ψ 4 ( x ). k= m ρ k(x k (t)) RBFNN W T S (Z ) n (3) s used to approxmate the unknown contnuous functon h (Z ). From (3) and (6), and usng the nequalty δ (Z )s 4 s ε, W T S (Z )s

5 98 Internatonal Journal of Automaton and Computng (3), June 3 λ η (Z )S (Z )s η,weobtan V s V w V U s g (t)u λ η (Z )S (Z )s 4 s m ε eγτmax ( τ ( s max) c ) ρ k(x k (t)) η k= c3 z ( s 4λ c )ψ( x ) 4 a p b p 4 γ VU. (7) Consder the followng control law u (t) =N(ς )(k s ˆλ η (Z )S (Z )s ) (8) ς = k s ˆλ η (Z )S (Z )s (9) where k > s a postve constant, specfed by the desgner. 4 The adaptaton law s gven by ˆλ = ω η (Z )S (Z )s σ ˆλ () where η >, σ >, and ω >. Consder the followng Lyapunov functon canddate V = V s V w V U λ. () ω Dfferentatng () wth respect to t yelds V s g (t)u λ η (Z )S (Z )s 4 s λ ˆλ ω m ε eγτmax ( τ ( s max) c ) ρ k(x k (t)) η c3 z 4λ p k= ( s c )ψ( x ) 4 p 4 γ VU. () Substtutng (8) and (9) nto (), and utlzng the nequalty σ λˆλ ω σ λ ω σ ω λ yelds V (k 4 )s σ λ ω g (t)n(ς ) ς ς σ m λ eγτmax ω ( τ ( s ) ρ max) c k(x k (t)) η ε c3 z 4λ a p k= ( s )ψ c ( x ) 4 b p 4 γ VU. (3) Remark. It s clearly seen from (3) that usng approprate Lyapunov-Krasovsk functonals not only effectvely compensates for the delayed states, but also relaxes the assumpton of the upper bound functons of tme-varyng delayed uncertantes. These upper bound functons are unknown contnuous functons n ths paper, whle they must be known n [6 8,,4,5]. Moreover, the controller u does not requre the delay ntegral term. Ths sgnfcantly allevates the computatonal burden. Defne a compact set as Ω z = {[x T,s,γ,β ] x Ω, =,,m, x d Ω d } Ω z = {[x T,s,γ,β,u,,u ] T x Ω, =,,m, x d Ω d, =,,} Ω c = {x s c, x d Ω d }, =,,m where Ω R n s a suffcently large compact set, and c s a postve constant that can be chosen arbtrarly small. Theorem. Consder the closed-loop system consstng of the plant () under Assumptons 7, and the adaptve control gven by (8) (). For bounded ntal condtons, all the sgnals n the closed-loop system are sem-globally unformly ultmately bounded, the parameter estmates ˆλ {ˆλ λ w μ } x {x x x d c ( Λ ) w () ( Λ )c [ ]max{ b μ,c }} Ω. λ Proof. Gven the Lyapunov functon canddate of (), the proof ncludes the followng three cases. Case. s k c k, k =,,,m. Because s k c k, k =,,,m and ρ k (x k (t)) are contnuous functons, we can let ρ,max = ψ,max = From (3), we obtan k= m max ρ k(x k (t)) s k c k max s k c k k =,,m ψ 4 ( x ). V (k 4 )s γ VU σ λ g (t)n(ς ) ς ω c 3 z ς μ (4) 4λ where μ = eγτmax ( τ max) ρ,max η ε b p 4 σ λ ψ, max. ω a p From () and (4), we obtan V (t) λ V (t)μ g (t)n(ς ) ς ς (5) where λ =mn{(k.5)b, γ, σ, c 3 4c }. From (5), we obtan V (t) C e λ t (g (τ)n(ς )) ς e λ τ dτ (6)

6 X. C. Sh and T. P. Zhang / Adaptve Trackng Control of Uncertan MIMO Nonlnear Systems 99 where C = μ λ V (). Accordng to Lemma, we have V (t), ς (t) and t g(τ)n(ς) ςdτ are bounded on [,t f ]. Therefore, s and λ are bounded on [,t f ]. Let C ς be the upper bound of e λ t t (g(τ)n(ς)) ςdτ on [, ), μ = C Cς, then λ μ ω and s b μ. Defne w =[e,e,,e n ] T R n. From (), we know that: ) There s a state space representaton for mappng s =[Λ T ]e,.e., ẇ = A s w b s s wth Λ = [λ,λ,,λ,n ] T,b s = [,,, ] T R n,a s beng a stable matrx. ) There s a postve constant c such that e As t c e λt.3)thesolutonforw s w (t) =e As w () Accordngly, t follows that e As (t τ )b s s (τ)dτ. w (t) c w e λt c e λ(t τ) s (τ) dτ. Let μ =max{ b μ,c }. Therefore, we have c μ w (t) c w (). (7) λ Notng s =Λ T w e n,e = [ ] T,weobtan w T,e n e w e n ( Λ ) w s. Substtutng (7) nto the above nequalty leads to ( Λ )c e c ( Λ ) w () [ ] μ. λ Notng x = e x d and Assumpton, we obtan x e x d c ( Λ ) w () x d ( Λ )c [ ]max{ b μ,c } L. λ Therefore, we can conclude from Case that all the closedloop sgnals are sem-globally unformly ultmately bounded for bounded ntal condtons. Case. s >c. From (4), we obtan where μ = η V (k 4 )s γ VU σ λ ω g (t)n(ς ) ς ς μ (8) a p b p 4 ε σ ω λ.smlarly, we have V (t) μ, λ μ ω and s b μ. Case 3. Let Σ I = { : s c, =,,m} and Σ J = { : s > c, =,,m}. From Case, for Σ J,wehave s b μ.therefove,let ρ,max = k Σ I k Σ J ψ,max =max x Ω ψ( x ) max ρ k(x k (t)) s k c k max ρ k(x k (t)) (9) s k b k μ k where Ω={x =[x T,,x T m] T s c, Σ I, c < s b μ, Σ J}. From (3) and (9), we can also obtan (4). Thus, smlarly, as dscussed n Case, we can conclude that Theorem holds. 4 Smulaton results To verfy the effectveness of the proposed approach, a numercal smulaton s demonstrated. Example. Consder the followng MIMO nonlnear system wth unmodeled dynamcs and tme-varyng delays: ż = z.5x sn(t) ẋ = x ẋ = x.3sn(x ).x (t τ (t)) ẋ = x ( sn (x ))u.8(x x x )cos(.5t)z. sn(t) ẋ = x u (x x.5cos(x ))u y = x.x (t τ (t)) sn(x (t τ (t))) (3 sn(x ))u.5(x x )sn(t) z.cos(.5x ) y = x (3) where u and u are the control nputs. The control obectve s to make the system outputs y and y follow the desred traectores y d and y d. Neural networks Ŵ T S(Z ) and Ŵ T S(Z ) contan 5 and 6 nodes wth centers taken randomly n the ntervals [.5,.5], the wdths are and 4. The desgn parameters of the above controller are k =.5, k =.8, c = c =., η = 8, η = 5, σ =., σ =., ω = 8, ω =, τ max =, and τ max =.5. The ntal condton s [x (),x (),x (),x (),z()] T =[,,,, ] T, tme delays are τ (t) =.( sn(t)), and τ (t) =.5cos(t), [ˆλ (), ˆλ ()] T =[, ] T, [ς (),ς ()] T =[, ] T. Choose the desred trackng traectores as y d =.5[sn(t)sn(.5t)],y d =sn(.5t).5sn(.5t). The smulaton results are shown n Fgs. 9. The smulaton results show that all the sgnals n the closed-loop control system are bounded and the approach of the proposed adaptve control s effectve. Furthermore, from Fgs. 4, t can be seen that farly good trackng performance s obtaned.

7 Internatonal Journal of Automaton and Computng (3), June 3 Fg. Output y and desred traectory y d Fg. 6 Control sgnal u Fg. Output y and desred traectory y d Fg. 7 Unmodeled dynamcs z Fg. 3 Trackng error e Fg. 8 Nussbaum tunng parameters ζ and ζ Fg. 9 Adaptve adustng parameters ˆλ and ˆλ Fg. 4 Trackng error e Fg. 5 Control sgnal u 5 Conclusons Adaptve neural trackng control has been proposed for a class of MIMO tme-varyng delay nonlnear systems wth completely unknown vrtual control drectons, unknown uncertan dynamc dsturbances, and unmodeled dynamcs. RBFNNs have been employed to approxmate unknown contnuous functons. The restrctons of unmodeled dynamcs are relaxed by utlzng Young s nequalty. Based on Lyapunov theory, the closed-loop system s proved to be sem-globally unformly ultmately bounded. In the future work, we wll further relax the restrcton of the tmevaryng delays uncertantes and extend the adaptve neural

8 X. C. Sh and T. P. Zhang / Adaptve Trackng Control of Uncertan MIMO Nonlnear Systems trackng control to a class of MIMO stochastc nonlnear systems. References [] S. S. Ge, C. C. Huang, T. Zhang. Adaptve neural network control of nonlnear systems by state and output feedback. IEEE Transactons on Systems, Man, and Cybernetcs Part B: Cybernetcs, vol. 9, no. 6, pp , 999. [] S. S. Ge, C. Wang, T. H. Lee. Adaptve backsteppng control of a class of chaotc systems. Internatonal Journal of Bfurcaton and Chaos, vol., no. 5, pp ,. [3] S. S. Ge, C. C. Huang, T. Zhang. Nonlnear adaptve control usng neural networks and ts applcaton to CSTR systems. Journal of Process Control, vol. 9, no. 4, pp , 999. [4] T. Zhang, S. S. Ge, C. C. Hang. Adaptve neural network control for strct-feedback nonlnear systems usng backsteppng desgn. Automatca, vol. 36, no., pp ,. [5] I. Mzumoto, T. W Chen, S. Ohdara, K. Makoto, I. Zenta. Adaptve output feedback control of general MIMO systems usng multrate samplng and ts applcaton to a cart-crane system. Automatca, vol. 43, no., pp , 7. [6] T. P. Zhang, S. S. Ge. Adaptve neural control of MIMO nonlnear state tme-varyng delay systems wth unknown dead-zones and gan sgns. Automatca, vol. 43, no. 6, pp. 33, 7. [7] H. B. Qan, T. P. Zhang. Adaptve control of MIMO nonlnear tme-varyng delay systems wth unknown gan sgns. Control and Decson, vol. 3, no., pp , 8. (n Chnese) [8] S.S.Ge,F.Hong,T.H.Lee.Adaptveneuralcontrolof nonlnear tme-delay systems wth unknown vrtual control coeffcents. IEEE Transactons on Systems, Man, and Cybernetcs Part B: Cybernetcs, vol. 34, no., pp , 4. [9] J. Hale. Theory of Functonal Dfferental Equatons, New York: Sprnger, 977. [] M. Jankovc. Control Lyapunov-Razumkhn functons and robust stablzaton of tme delay systems. IEEE Transactons on Automatc Control, vol. 46, no. 7, pp. 48 6,. [] S. S. Ge, F. Hong, T. H. Lee. Robust adaptve control of nonlnear systems wth unknown tme delays. Automatca, vol. 4, no. 7, pp. 8 9, 5. [] S. S. Ge, K. P. Tee. Approxmaton-based control of nonlnear MIMO tme-delay systems. Automatca, vol. 43, no., pp. 3 43, 7. [3] M. Wang, B. Chen, K. F. Lu, S. Y. Zhang. Adaptve fuzzy trackng control of nonlnear tme-delay systems wth unknown vrtual control coeffcents. Informaton Scences, vol. 78, no., pp , 8. [4] Q. Q. Zhu, T. P. Zhang, B. C. Zhu. Adaptve control of MIMO nonlnear systems wth dsturbances and tme delays. In Proceedngs of the Chnese Control and Decson Conference, IEEE, Xuzhou, Chna, pp. 99 4,. [5] T. P. Zhang, C. Y. Zhou, Q. Zhu. Adaptve varable structure control of MIMO nonlnear systems wth tme-varyng delays and unknown dead-zones. Internatonal Journal of Automaton and Computng, vol. 6, no., pp. 4 36, 9. [6] H. Y. Yue, J. M. L. Adaptve fuzzy dynamc surface control for a class of perturbed nonlnear tme-varyng delay systems wth unknown dead-zone. Internatonal Journal of Automaton and Computng, vol. 9, no. 5, pp ,. [7] D. L, J. M. L. Adaptve teratve learnng control for nonlnearly parameterzed systems wth unknown tme-varyng delay and unknown control drecton. Internatonal Journal of Automaton and Computng, vol. 9, no. 6, pp ,. [8] S. C. Tong, X. L. He, Y. M. L, H. G. Zhang. Adaptve fuzzy backsteppng robust control for uncertan nonlnear systems based on small-gan approach. Fuzzy Sets and Systems, vol. 6, no. 6, pp ,. [9] S. C. Tong, Y. M. L. Fuzzy adaptve robust backsteppng stablzaton for SISO nonlnear systems wth unknown vrtual control drecton. Informaton Scences, vol. 8, no. 3, pp ,. [] T. P. Zhang, Y. Lu. Adaptve dynamc surface control of nonlnear systems wth unmodeled dynamcs. Control and Decson, vol. 7, no. 3, pp ,. (n Chnese) [] Z. P. Jang, D. J. Hll. A robust adaptve backsteppng scheme for nonlnear systems wth unmodeled dynamcs. IEEE Transactons on Automatc Control, vol. 44, no. 9, pp. 75-7, 999. [] M. Krstc, I. Kanellakopoulos, P. V. Kokotovc. Nonlnear and Adaptve Control Desgn, New York: Wley, 995. [3] R. M. Sanner, J. J. E. Slotne. Gaussan networks for drect adaptve control. IEEE Transactons on Neural Networks, vol. 3, no. 6, pp , 99. [4] S. S. Ge, C. C. Hang, T. H. Lee, T. Zhang. Stable Adaptve Neural Network Control, Boston: Kluwer Academc, pp. 7 46,. Xao-Cheng Sh receved the B. Eng. degree n automaton from Yangzhou Unversty, Chna n. She s currently a master student n control theory and control engneerng at Yangzhou Unversty, Chna. Her research nterests nclude robust adaptve control, dynamc surface control, and nonlnear control. E-mal: xaocheng sh@63.com Tan-Png Zhang receved the B. Sc. degree n mathematcs from Yangzhou Teachers College, Yangzhou, Chna n 986, the M. Sc. degree n operatons research and control theory from East Chna Normal Unversty, Shangha, Chna n 99, and the Ph. D. degree n automatc control theory and applcatons from Southeast Unversty, Nanng, Chna n 996. He s now a professor n the College of Informaton Engneerng, Yangzhou Unversty, Yangzhou, Chna. From 5 to 6, he was a vstng scentst n the Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore, Sngapore. He has publshed more than 8 papers n ournals and conferences. Hs research nterests nclude fuzzy control, adaptve control, ntellgent control, and nonlnear control. E-mal: tpzhang@yzu.edu.cn (Correspondng author)